You are correct - it registers negative height.
Above sea level (or above mean sea level) is your height with reference to sea level which is, apart from tides, swells, etc the same everywhere averaged out. Above ground level is measured from your current location. The quad starts with a height of zero (ground level) and displays its height above this datum. I have never flown downhill, but I assume that if you did, your height would register as negative. I sit to be corrected of course.
Aviation maps show high obstacles measured both ways, above local ground height and above mean sea level.